I am not primarily a classical performer but rhythm is a very vital part of my own music. I think lostinidlewonder is correct in that if you develop the ability to actually feel independent, simultaneous rhythms then you have cracked the general problem. Counting and other devices only provide expedient special solutions to special situations. A lot depends on whether any of the rhythms have aligned notes and whether the precise juxtaposed rhythms are vital to the music. Often things like five against six are just groups of regularly spaced notes which the composer has notated that way by convention. In other words the precise rhythmic feel of five against six is not intrinsic to the required sound and in any case, each stream of notes is quite regular.
The real difficulty arises when both these conditions are false, i.e. the juxtaposed rhythmic feeling is vital and the notes are not evenly spaced, even perhaps wholly unaligned. A very simple example of this is the little section of eight bars in Db in the Rhapsody In Blue (page 18 in my copy), where one of the main themes, itself containing syncopation and triplets, is played over a sort of inverted stride bass. There are three ways of aligning it and Gershwin uses two in succession - very effective in fact. When I learned this piece with my teacher many years ago I tried counting various ways and, as he rightly pointed out with much laughter, the whole thing started sounding like Gilbert and Sullivan gone wrong (6/8) which isn't the intended effect at all. He insisted I work until I could feel left and right hand rhythms at once at a reasonably slow tempo. I think that was my first time to do that, and it took me ages. However, with time, the general knack can be acquired and applied to any new situation.